$n$-维分段光滑微分系统的周期轨分支

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 1043-1052.

$n$-维分段光滑微分系统的周期轨分支

杨纪华^1,*(),张二丽²

¹ 宁夏师范学院数学与计算机科学学院宁夏固原 756000
² 郑州财经学院信息工程学院郑州 450001

收稿日期:2018-11-09 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 杨纪华 E-mail:jihua1113@163.com
基金资助:
宁夏高等学校科学技术研究项目(NGY2020074);国家自然科学基金(11701306);宁夏自然科学基金(2019AAC03247);宁夏高等学校一流学科建设（教育学学科）(NXYLXK2017B11);河南省高等学校青年骨干教师培养计划(2017GGJS202);河南省高等学校青年骨干教师培养计划(2016GGJS-190);河南省高等学校重点科研项目(19A110033);河南省高等学校重点科研项目(19B110014)

Bifurcation of Periodic Orbits of an n-Dimensional Piecewise Smooth Differential System

Jihua Yang^1,*(),Erli Zhang²

¹ School of Mathematics and Computer Science, Ningxia Normal University, Ningxia Guyuan 756000
² School of Information Engineering, Zhengzhou Institute of Finance and Economics, Zhengzhou 450001

Received:2018-11-09 Online:2020-08-26 Published:2020-08-20
Contact: Jihua Yang E-mail:jihua1113@163.com
Supported by:
the Higher Education Science and Technology Program of Ningxia(NGY2020074);the NSFC(11701306);the NSF of Ningxia(2019AAC03247);the Construction of First-class Disciplines of Higher Education of Ningxia (pedagogy)(NXYLXK2017B11);the Training Plan for Young Scholar of Higher Education of Henan Province(2017GGJS202);the Training Plan for Young Scholar of Higher Education of Henan Province(2016GGJS-190);the Key Program of Higher Education of Henan Province(19A110033);the Key Program of Higher Education of Henan Province(19B110014)

摘要/Abstract

摘要：

研究了$n$ -维分段光滑扰动微分系统

$ \begin{eqnarray*} \ \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^+_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^+_2({\bf x}),\\ \dot{x}_3=\varepsilon g^+_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^+_n({\bf x}),\\ \end{array}\right.x_1\geq0,\quad \quad \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^-_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^-_2({\bf x}),\\ \dot{x}_3=\varepsilon g^-_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^-_n({\bf x}),\\ \end{array}\right.x_1<0, \end{eqnarray*} $

其中${\bf x}=(x_1,x_2,\cdots,x_n)^T$, $0<\varepsilon\ll1$,且$g^\pm_i({\bf x})$, $i=1,2,\cdots,n$是关于${\bf x}$的$m$次实系数多项式.应用一阶Melnikov向量函数，得到了从其未扰动系统分支出周期轨个数的上界.

关键词: n-维分段光滑微分系统, 周期轨, Melnikov向量函数

Abstract:

In this paper, we study the following $n$-dimensional piecewise smooth differential system

where ${\bf x}=(x_1,x_2,\cdots,x_n)^T$, $0<\varepsilon\ll1$, and $g^\pm_i({\bf x})$, $i=1,2,\cdots,n$ are real polynomials of ${\bf x}$ with degree $m$. By using the first order Melnikov vector function, we obtain the upper bound of the number of periodic orbits bifurcating from the unperturbed system.

Key words: n-Dimensional piecewise smooth differential system, Periodic orbit, Melnikov vector function

中图分类号:

O175.12

杨纪华,张二丽. $n$-维分段光滑微分系统的周期轨分支[J]. 数学物理学报, 2020, 40(4): 1043-1052.

Jihua Yang,Erli Zhang. Bifurcation of Periodic Orbits of an n-Dimensional Piecewise Smooth Differential System[J]. Acta mathematica scientia,Series A, 2020, 40(4): 1043-1052.

参考文献

1	Liu X , Han M . Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Internat J Bifur Chaos Appl Sci Engrg, 2010, 20, 1379- 1390
2	Liang F , Han M , Romanovski V . Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop. Nonlinear Anal, 2012, 75, 4355- 4374
3	Xiong Y . Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters. Nonlinear Analysis: Real World Applications, 2018, 41, 384- 400
4	Li S , Liu C . A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system. J Math Anal Appl, 2015, 428, 1354- 1367
5	Yang J , Zhao L . Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems Series B, 2017, 22, 2417- 2425
6	Han M , Sheng L . Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J Appl Anal Comput, 2015, 5, 809- 815
7	Llibre J , Mereu A , Novaes D . Averaging theory for discontinuous piecewise differential systems. J Differential Equations, 2015, 258, 4007- 4032
8	Han M . On the maximum number of periodic solutions of piecewise smooth periodic equations by average method. J Appl Anal Comput, 2017, 7, 788- 794
9	Yang J , Zhao L . Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J Differential Equations, 2018, 264, 5734- 5757
10	Llibre J , Makhlouf A . Bifurcation of limit cycles from a two-dimensional center inside $\mathbb{R} ^n$. Nonlinear Analysis, 2010, 72, 1387- 1392
11	Barreira L , Llibre J , Valls C . Bifurcation of limit cycles from a 4-dimensional center in $\mathbb{R} ^m$ in resonance $1:N$. J Math Anal Appl, 2012, 389, 754- 768
12	Llibre J , Teixeira M , Zeli I . Birth of limit cycles for a class of continuous and discontinuous differential systems in $(d+2)$-dimension. Dynamical Systems, 2016, 31 (3): 237- 250
13	Tian H , Han M . Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems. J Differential Equations, 2017, 263, 7448- 7474
14	Han M , Sun H , Balanov Z . Upper estimates for the number of periodic solutions to multi-dimensional systems. J Differential Equations, 2019, 266, 8281- 8293